Kerr-lens mode locking (KLM)

Kerr-lens mode locking (KLM) is a form of passive mode locking that uses the optical Kerr effect in the laser crystal itself. The refractive index of the crystal rises with intensity, $n = n_0 + n_2 I$, and a beam that is most intense on its axis therefore sees a positive lens whose strength grows with power. A pulse, with its high peak power, is focused more tightly than continuous light at the same average power. When the cavity is arranged so that the more tightly focused mode suffers less loss or sees more gain, the laser prefers to run as a train of short pulses. KLM was discovered in 1991 in Ti:sapphire, when Spence, Kean and Sibbett reported 60 fs pulses from a laser with no separate mode-locking element, and it remains the method by which Ti:sapphire oscillators produce pulses of 10–100 fs at repetition rates near 80 MHz.

Self-focusing as a fast saturable absorber

A saturable absorber transmits more at high intensity. The Kerr lens, combined with an aperture, does the same: the effective loss falls as the peak power rises. Because the electronic Kerr response lasts well under 10 fs, this artificial absorber recovers essentially instantly and can shape pulses shorter than any real absorber could. For a Gaussian beam of radius $w$ and peak power $P$ passing a thin crystal of length $L$, the parabolic part of the nonlinear phase acts as a lens of focal length

$$f_{NL} = \frac{\pi w^4}{8\,n_2\,P\,L}.$$

For a typical oscillator with 5 W of intracavity average power at 80 MHz, each pulse carries 62.5 nJ, and at 50 fs with a sech² shape the peak power is about 1.1 MW. In sapphire at 800 nm, with $n_2 \approx 3 \times 10^{-20}$ m²/W and $n_0 = 1.76$, the critical power for catastrophic self-focusing is about 1.8 MW, so the pulse runs at about 0.6 of it. With a 20 µm beam radius the peak index change is about 5 × 10⁻⁵, and the formula above gives $f_{NL} \approx 0.6$ mm for a 3 mm crystal. A focal length shorter than the crystal and comparable to the 2.8 mm Rayleigh range means the thin-lens picture has broken down; the Kerr lens is a strong perturbation of the cavity mode, and its effect is calculated by propagating the beam through the crystal in slices.

Hard and soft apertures

In hard-aperture KLM a slit or iris is placed where the pulsed mode is smaller than the continuous-wave mode, typically near an end mirror; the pulse passes with less clipping loss. In soft-aperture KLM there is no physical aperture: the pump beam in the crystal acts as one, since a tighter laser mode overlaps better with the pump and sees higher gain. Most commercial oscillators use soft-aperture KLM, often with a slit only to suppress continuous-wave lasing.

Both schemes need the cavity near the edge of a stability zone, where a small change of lens power produces a large change in mode size. The curved-mirror separation around the crystal is adjusted so that the Kerr lens pushes the pulsed mode in the favourable direction. The modulation of loss obtained in this way is typically a few percent.

Pulse shaping and the few-cycle limit

The Kerr effect also produces self-phase modulation, which chirps the pulse. Net negative round-trip dispersion, supplied by a prism pair or chirped mirrors (see pulse compressor), balances it in a way similar to an optical soliton, and the steady-state pulse duration is set by this balance and by the gain bandwidth. With carefully designed double-chirped mirrors, Ti:sapphire oscillators have produced pulses of about 5 fs directly, fewer than two optical cycles at 800 nm, where one cycle lasts 2.67 fs. At that duration the phase of the carrier under the envelope becomes significant, which is the subject of the carrier-envelope offset entry.

Starting and practical issues

KLM is often not self-starting: the Kerr lens is negligible for the low-intensity noise of continuous-wave operation, so the laser needs a perturbation, such as tapping a mirror or moving a prism, to create an initial intensity spike. Adding a SESAM or operating closer to the stability edge helps start it. Other pitfalls are multiple-pulse operation at high intracavity power, continuous-wave breakthrough alongside the pulse, and sensitivity to the pump-beam pointing and to air currents in the cavity. The same principle is used in Yb-doped bulk and thin-disk oscillators, in which the Kerr medium is sometimes a separate plate.

Common questions

Is KLM active or passive mode locking?

Passive: no modulator is driven, and the pulse modulates its own loss through the Kerr lens.

Why is Ti:sapphire so well suited to KLM?

Its gain bandwidth supports pulses of a few femtoseconds, and the pump beam is focused tightly in a crystal of a few millimetres, which gives the intensity needed for a strong Kerr lens at nanojoule pulse energies.

How short can a KLM laser make pulses?

About 5 fs from Ti:sapphire oscillators, limited by the gain bandwidth and by how accurately the mirrors compensate dispersion across it. Shorter pulses are obtained by broadening the spectrum outside the laser.

References: D. E. Spence, P. N. Kean, W. Sibbett, Opt. Lett. 16, 42 (1991); R. Ell et al., Opt. Lett. 26, 373 (2001); H. A. Haus, IEEE J. Sel. Top. Quantum Electron. 6, 1173 (2000); J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, 2006); A. E. Siegman, Lasers (University Science Books, 1986).